The boundary conditions specify the value of electric & magnetic fields or their derivatives or the products with some directional vectors on the boundary of the computation domain to find a particular solution. Three types of boundary conditions are primarily needed to be considered during the numerical computation. These are- 1) Perfect Electric Conductor 2) Perfect Magnetic Conductor 3) Perfectly Matched Layer.
The interior boundaries of the cross-sectional geometry of the PCF are set at continuity boundary conditions. The outermost boundary of the cross-section is set at perfect electric conductor boundary condition when solving for magnetic field intensity, H. Perfect magnetic conductor boundary condition is used when solving for electric field, E. This is a valid boundary condition as the radius of the geometry is chosen large enough for the electric or magnetic field to decay to zero.
2.4.1 Perfect Electric Conductor:
This boundary condition can be expressed as,
n B 0 (2.6) nE 0 (2.7) Equation (2.7) sets the tangential component of the electric field to zero at the boundary
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Here, 𝑛̂ is the unit normal vector to the boundary. According to this condition, tangential components of 𝐸⃗ and normal components of 𝐵⃗ is continuous across any interface.
2.4.2 Perfect Magnetic Conductor:
In this case, the boundary condition is expressed as-
nD 0 (2.8) nH 0 (2.9) Equatio (9) sets the tangential component of the magnetic field to zero at the boundary In this case, the tangential components of 𝐻⃗⃗ and normal components of 𝐷⃗⃗ is continuous across any interface.
2.4.3 Continuity:
The continuity boundary condition,
n(H1H2)0 (2.10)
n(E1E2) 0 (2.11) is the natural boundary condition ensuring continuity of the tangential components of the
electric and magnetic fields.
2.4.4 Perfectly Matched Layer:
Numerical simulation of propagating waves in unbounded spatial domains is a challenge common to many branches of engineering and applied mathematics. Perfectly matched layers (PML) are a novel technique for simulating the absorption of waves in open domains. The equations modeling the dynamics of phenomena of interest are usually posed as differential equations or integral equations which must be solved at every time instant. In many application areas like general relativity and acoustics, the underlying equations are systems of second order hyperbolic partial differential equations. In numerical treatment of such problems, the equations are often rewritten as first order systems and are solved in this form.
For this reason, many existing PML models have been developed.
A perfectly matched layer is an artificial boundary condition implying perfect absorption of incident field. This boundary condition is required for approximating infinite zone beyond the waveguide outer edge to a finite domain of numerical analysis. Effect of PML on
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numerical solutions obtained will be more prominent when confinement of field in the PCF is weak. This layer can also be utilized to find out the complex part of effective index.
There are several different PML formulations. However, all PML’s essentially act as a lossy material. The lossy material, or lossy layer, is used to absorb the fields traveling away from the interior of the grid. The PML is anisotropic and constructed in such a way that there is no loss in the direction tangential to the interface between the lossless region and the PML.
However, in the PML there is always loss in the direction normal to the interface.
The PML was originally proposed by J. P. Berenger in 1994[30]. In that original work he split each field component into two separate parts. The actual field components were the sum of these two parts but by splitting the field Berenger could create an (non-physical) anisotropic medium with the necessary phase velocity and conductivity to eliminate reflections at an interface between a PML and non-PML region.
The PML region can be viewed as a perfect absorber with a certain magnitude of conductivity. However, the optimized conductivity is calculated from certain sets of equations. In our work, we have considered cylindrical PML available in the commercial software.
Fig. 2.2 PML region surrounding the waveguide structure.
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Here, e is the thickness of the PML layer which is ideally a multiple of the operating wavelength.
The wave equations in the PML can be done after doing rigorous analysis-
H j n sE
2(2.12)
E j
0sH(2.13)
20 0
1 e 1 m
s j j
n
(2.14
) Here,E: Electric field H: Magnetic Field
𝜎e: Electric conductivity of PML 𝜎m: Magnetic conductivity of PML
To avoid numerical reflection, conductivity in the PML region is graded to a peak value rather than an abrupt rise as shown in Fig. 2.5.
Fig. 2.3: Grading of PML conductivity
24 In this case, the PML parameters become-
2
1 rin
s jk e
in PML region
=1 in other regions
(2.15) Where k is defined for wavelength by3 1 4 e ln
k n R
(2.16)
Where R is the reflection coefficient of electromagnetic field from the interface to be minimized.
max
0 0 0
exp[( 2 d( ) )m ]
R dp
n n d
(2.17) For a perfectly matched condition to secure zero reflection from the interface, we can write- 20
e m
n m
(2.18) For this maximum conductivity is defined as below,
max 1 0 ln(1) 2
m cn
d R
(2.19) Here, m is the order of polynomial for grading conductivity. These equations imply that, minimum reflection will occur for a maximum conductivity. However, numerical error terms imply that there is an optimum magnitude of reflection for accurate propagation constant. A stable value of complex effective index can be obtained by altering the thickness of PML and distance of PML from center of the PCF.
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Chapter 03
Some Properties of Optical Fiber
The introduction of optical fiber system revolutionized the communication network. The low transmission loss and the large bandwidth capability of the fiber systems allow signals to be transmitted for establishing communication contacts over large distances with few or no provisions of intermediate application. There are many properties of optical fiber such as dispersion, birefringence, effective material loss, confinement loss, bending loss, non- linearity etc. The most focused properties of fiber are described below: